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Manager  S
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in how many ways can 9 identical balls be distributed among four baske  [#permalink]

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Question Stats: 47% (02:04) correct 53% (02:14) wrong based on 88 sessions

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Q.In how many ways can 9 identical balls be distributed among four baskets such that each basket gets at least one ball?
A. 35
B. 56
C. 63
D. 70
E. 126
Please anybody solve this question and explain me how did you solve this question i am not able to do this question thanks in advance
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in how many ways can 9 identical balls be distributed among four baske  [#permalink]

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rishabhmishra wrote:
Q.In how many ways can 9 identical balls be distributed among four baskets such that each basket gets at least one ball?
A. 35
B. 56
C. 63
D. 70
E. 126
Please anybody solve this question and explain me how did you solve this question i am not able to do this question thanks in advance

This question is based on Distribution principle. It can be done in two ways

This is like find total Natural number solution of an equation a+b+c+d = 9

Method 1:

Natural Number solution of an equation in r variables = (n-1)C(r-1) = (9-1)C(4-1) = 8C3 = 56

Method 2:

Assign one ball to each basket i.e. 4 balls are gone and we are left with only 5 balls to assign to the baskets
Now every basket needs balls from 0 to 5
a+b+c+d = 5
and a, b, c and d may be any integer from 0 to 5
Now manually find solutions which is long but you get a pattern in that as well leading you to the answer i,e, 56

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Senior PS Moderator V
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Re: in how many ways can 9 identical balls be distributed among four baske  [#permalink]

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rishabhmishra wrote:
Q.In how many ways can 9 identical balls be distributed among four baskets such that each basket gets at least one ball?
A. 35
B. 56
C. 63
D. 70
E. 126
Please anybody solve this question and explain me how did you solve this question i am not able to do this question thanks in advance

If we were to manually list down all the possibilities for the four baskets(and the 4 digit number is the total number of balls in the basket number one, two, three, and four)

6-1-1-1 (4 ways) - 6111,1611,1161,1116
3-2-2-2 (4 ways) - 3222,2322,2232,2223
4-2-2-1 (12 ways) - 4221,4212,4122,2124,2142,2421,2412,2241,2214,1224,1242,1422
4-3-1-1 (12 ways) - 4311,4131,4113,3411,3114,3141,1341,1431,1143,1134,1413,1314
1-2-3-3 (12 ways) - 1233,1323,1332,2133,2331,2313,3321,3231,3123,3132,3312,3213
5-1-1-2 (12 ways) - 5112,5121,5211,2511,2151,2115,1215,1251,1125,1152,1521,1512

Therefore, there are $$4*2 + 12*4$$ or 56 ways(Option B) in which the 9 balls can be arranged among the 4 baskets.
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Math Expert V
Joined: 02 Aug 2009
Posts: 7958
Re: in how many ways can 9 identical balls be distributed among four baske  [#permalink]

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rishabhmishra wrote:
Q.In how many ways can 9 identical balls be distributed among four baskets such that each basket gets at least one ball?
A. 35
B. 56
C. 63
D. 70
E. 126
Please anybody solve this question and explain me how did you solve this question i am not able to do this question thanks in advance

hi..

the question does NOT mention about the baskets, if they too are identical..
1) Identical boxes..
a) 6,1,1,1
b) 5,2,1,1
c) 4,3,1,1
d) 4,2,2,1
e) 3,3,2,1
f) 3,2,2,2
so 6 ways

2) non identical boxes..
a) 6,1,1,1 ............. $$\frac{4!}{3!} = 4$$
b) 5,2,1,1............. $$\frac{4!}{2!} = 4*3=12$$
c) 4,3,1,1............. $$\frac{4!}{2!} = 4*3=12$$
d) 4,2,2,1............. $$\frac{4!}{2!} = 4*3=12$$
e) 3,3,2,1............. $$\frac{4!}{2!} = 4*3=12$$
f) 3,2,2,2............. $$\frac{4!}{3!} = 4$$
so 4+12+12+12+12+4=56
In these cases we can take it as an equation a+b+c+d=9, as shown above..

For other type of such possible scenarios
https://gmatclub.com/forum/topic215915.html
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Re: in how many ways can 9 identical balls be distributed among four baske  [#permalink]

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chetan2u

If we remove the last condition i.e. such that each basket gets at least one ball. The possible combination - 9!^4 am I correct?
Intern  B
Joined: 27 May 2019
Posts: 1
Re: in how many ways can 9 identical balls be distributed among four baske  [#permalink]

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GMATinsight wrote:
rishabhmishra wrote:
Q.In how many ways can 9 identical balls be distributed among four baskets such that each basket gets at least one ball?
A. 35
B. 56
C. 63
D. 70
E. 126
Please anybody solve this question and explain me how did you solve this question i am not able to do this question thanks in advance

This question is based on Distribution principle. It can be done in two ways

This is like find total Natural number solution of an equation a+b+c+d = 9

Method 1:

Natural Number solution of an equation in r variables = (n-1)C(r-1) = (9-1)C(4-1) = 8C3 = 56

Method 2:

Assign one ball to each basket i.e. 4 balls are gone and we are left with only 5 balls to assign to the baskets
Now every basket needs balls from 0 to 5
a+b+c+d = 5
and a, b, c and d may be any integer from 0 to 5
Now manually find solutions which is long but you get a pattern in that as well leading you to the answer i,e, 56

Could you please explain what "natural number solution" means?

How would the approach change if the numbers changed?

e.g. if 10 identical balls were distributed among 4 boxes and each box gets at least 2 balls, in how many ways could the balls be distributed?

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Intern  B
Joined: 28 Jan 2019
Posts: 6
Location: India
Re: in how many ways can 9 identical balls be distributed among four baske  [#permalink]

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rishabhmishra wrote:
Q.In how many ways can 9 identical balls be distributed among four baskets such that each basket gets at least one ball?
A. 35
B. 56
C. 63
D. 70
E. 126
Please anybody solve this question and explain me how did you solve this question i am not able to do this question thanks in advance

As Baskets are 4, solution should be a multiple of 4. Instead of calculating it the long way, we can use Triage in this question.

Only solution multiple of 4 is B. Hence the Answer.
Intern  B
Joined: 10 Jul 2019
Posts: 4
Re: in how many ways can 9 identical balls be distributed among four baske  [#permalink]

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Q.In how many ways can 9 identical balls be distributed among four baskets such that each basket gets at least one ball?
A. 35
B. 56
C. 63
D. 70
E. 126

Correct Answere is B
Senior Manager  P
Joined: 15 Feb 2018
Posts: 365
Re: in how many ways can 9 identical balls be distributed among four baske  [#permalink]

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Anuragjn reasoning seems wrong to me, as we are summing different numbers of 4 baskets (not multiplying), so not necessarily divisible by 4

Can someone please explain?
Is this question within scope of gmat? Re: in how many ways can 9 identical balls be distributed among four baske   [#permalink] 18 Aug 2019, 23:07
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